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Amy Ferguson Moncure

IB AA · Unit 2 · 18B Trigonometric Functions

Related Angles: Cosine

Turn the angle until it matches 155° or 25°. Cosine is even, and a half turn changes the sign.

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Key Ideas Easy Medium Hard

cos(−θ) = cos θ

cos(180° − θ) = −cos θ

sin(90° + θ) = cos θ

What to Remember

Same cosine

cos(−θ) = cos θ

cos(θ + 360°) = cos θ

cos(360° − θ) = cos θ

Opposite cosine

cos(180° − θ) = −cos θ

cos(θ + 180°) = −cos θ

Sine of a related angle

sin(90° + θ) = cos θ

sin(θ − 90°) = −cos θ

Easy

Same value

Add full turns, or use cos(−θ) = cos θ.

Example
Given that cos 155° = −0.906, find cos 515°.

515° = 155° + 360°

cos(θ + 360°) = cos θ

cos 515° = cos 155°

cos 515° = −0.906

Answer: −0.906

1
Given that cos 155° = −0.906, find cos(−155°).
Show Answer

cos(−θ) = cos θ

cos(−155°) = cos 155°

cos(−155°) = −0.906

Answer: −0.906

2
Given that cos 155° = −0.906, find cos 205°.
Show Answer

205° = 360° − 155°

cos(360° − θ) = cos θ

cos 205° = cos 155°

cos 205° = −0.906

Answer: −0.906

3
Given that cos 155° = −0.906, find cos 875°.
Show Answer

875° − 2 × 360° = 875° − 720°

875° − 720° = 155°

cos 875° = cos 155°

cos 875° = −0.906

Answer: −0.906

4
Given that cos 155° = −0.906, find cos(−205°).
Show Answer

−205° + 360° = 155°

cos(−205°) = cos 155°

cos(−205°) = −0.906

Answer: −0.906

5
Given that cos 155° = −0.906, find cos(−515°).
Show Answer

cos(−θ) = cos θ

cos(−515°) = cos 515°

515° = 155° + 360°

cos 515° = cos 155°

cos(−515°) = −0.906

Answer: −0.906

Medium

Change the sign

180° − θ and θ + 180° both give the opposite cosine.

Example
Given that cos 155° = −0.906, find cos 25°.

25° = 180° − 155°

cos(180° − θ) = −cos θ

cos 25° = −cos 155°

cos 25° = −(−0.906)

cos 25° = 0.906

Answer: 0.906

1
Given that cos 155° = −0.906, find cos 335°.
Show Answer

335° = 155° + 180°

cos(θ + 180°) = −cos θ

cos 335° = −cos 155°

cos 335° = −(−0.906)

cos 335° = 0.906

Answer: 0.906

2
Given that cos 155° = −0.906, find cos(−25°).
Show Answer

cos(−θ) = cos θ

cos(−25°) = cos 25°

25° = 180° − 155°

cos 25° = −cos 155°

cos(−25°) = 0.906

Answer: 0.906

3
Given that cos 155° = −0.906, find cos 385°.
Show Answer

385° = 25° + 360°

cos 385° = cos 25°

cos 25° = −cos 155°

cos 385° = 0.906

Answer: 0.906

4
Given that cos 155° = −0.906, find cos 745°.
Show Answer

745° − 2 × 360° = 745° − 720°

745° − 720° = 25°

cos 745° = cos 25°

cos 745° = 0.906

Answer: 0.906

5
Given that cos 155° = −0.906, find cos(−335°).
Show Answer

cos(−335°) = cos 335°

335° = 155° + 180°

cos 335° = −cos 155°

cos(−335°) = 0.906

Answer: 0.906

Hard

A sine that matches

sin(90° + θ) equals cos θ.

Example
Given that cos 155° = −0.906, find sin 245°.

245° = 90° + 155°

sin(90° + θ) = cos θ

sin 245° = cos 155°

sin 245° = −0.906

Answer: −0.906

1
Given that cos 155° = −0.906, find sin 65°.
Show Answer

65° = 155° − 90°

sin(θ − 90°) = −cos θ

sin 65° = −cos 155°

sin 65° = −(−0.906)

sin 65° = 0.906

Answer: 0.906

2
Given that cos 155° = −0.906, find sin(−65°).
Show Answer

sin(−θ) = −sin θ

sin(−65°) = −sin 65°

sin 65° = 0.906

sin(−65°) = −0.906

Answer: −0.906

3
Given that cos 155° = −0.906, find sin 295°.
Show Answer

295° = 360° − 65°

sin(360° − α) = −sin α

sin 295° = −sin 65°

sin 295° = −0.906

Answer: −0.906

4
Given that cos 155° = −0.906, find sin 425°.
Show Answer

425° = 65° + 360°

sin 425° = sin 65°

sin 65° = 0.906

sin 425° = 0.906

Answer: 0.906

5
Given that cos 148° = −0.848, find sin 238°.
Show Answer

238° = 90° + 148°

sin(90° + θ) = cos θ

sin 238° = cos 148°

sin 238° = −0.848

Answer: −0.848

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